The convention in monetary economics has been to create monetary
aggregates by simply adding together the dollar amounts of the various
financial assets included in them. This is the simple-sum method of
aggregation. This procedure has been criticized because such monetary
aggregates are essentially indexes that weight each component financial
asset equally, a practice the is economically meaningful only under
special circumstances.

A number of alternative indexes of monetary aggregates have been
developed recently. The most well known are the Divisia monetary
aggregates developed by Barnett (1980). This article reviews the
theoretical basis for monetary aggregation and presents series of
Divisia monetary aggregates for an extended sample period. The behavior
of the simple-sum aggregates and their Divisia counterparts are compared
over this period.

THE THEORETICAL BASIS FOR

MONETARY AGGREGATION(1)

Simple-sum aggregation stemmed directly from the classical
economists' notion that the essential function of money is to
facilitate transactions, that is, to serve as a medium of exchange.
Assets that served as media of exchange were considered money and those
that did not, were not. By this definition only two assets, currency and
demand deposits, were considered money. Both assets were non-interest
bearing, and individuals were free to alter the composition of their
money holdings between currency and demand deposits at a fixed
one-to-one ratio. Consequently the monetary value of transactions was
exactly equal to the sum of the two monies.(2) Simple-sum aggregation
was a natural extension of both restricting the definition of money to
non-interest-bearing medium-of-exchange assets and of the fixed unitary exchange rate between the two alternative monies.(3)

In consumer demand theory, simple-sum aggregation is tantamount to
treating currency and demand deposits as if they are perfect
substitutes. Currency and demand deposits, however, are not equally
useful for all transactions, so this assumption was clearly
inappropriate. But, simple-sum aggregation of those two monetary assets
was still appropriate because the assets were non-interest bearing and
exchanged at a fixed one-to-one ratio. Consequently individuals would
allocate their portfolio of money between the two assets until they
equalized the marginal utilities of the last dollar held of each. Under
these conditions, simple-sum aggregation is appropriate if it is also
assumed that each agent is holding his equilibrium portfolio.

The recognition that non-interest-bearing demand deposits may have
paid an implicit interest weakened the theoretical justification for
simple-sum aggregation. A more serious blow to simple-sum aggregation,
however, was dealt by a shift in monetary theory to emphasizing the
store-of-value function of money.(4) That an asset could not be used
directly to facilitate transactions was no longer a sufficient condition
for excluding it from the definition of money. Instead, the asset
approach to money emphasized money's role as a temporary abode of
purchasing power that bridges the gap between the sale of one item and
the purchase of another. Currency and checking accounts are money
because they are both media of exchange and temporary abodes of
purchasing power. Non-medium of exchange assets are superior to currency
and non-interest-bearing checking accounts as stores of value because
they earn explicit interest. This superiority typically increases with
the length of time between the sale of one item and subsequent purchase
of another because the cost of getting into and out of such assets and
the medium of exchange assets is thought to be small and not
proportional to the size of the transaction.

This shift in emphasis in monetary theory dramatically expanded the
number of assets that were considered money and the number of
alternative monetary aggregates proliferated.(5) Nonetheless, the method
of aggregation remained the same - simple-sum aggregation.

As more financial assets came to be considered money, it became
increasingly clear that it was inappropriate to treat these assets as
perfect substitutes. Some financial assets have more
"moneyness" than others, and hence they should receive larger
weights. In what appears to be the first attempt at constructing a
theoretically preferable alternative to the simple-sum monetary
aggregate, Chetty (1969) added various savings-type deposits, weighted
by estimates of the degree of substitution between them and the pure
medium of exchange assets, to currency and demand deposits. Larger
weights were given to assets with a higher estimated degree of
substitution.(6)

Divisia aggregation, which also relies on consumer demand theory
and the theory of economic aggregation, treats monetary assets as
consumer durables such as cars, televisions and houses. They are held
for the flow of utility-generating monetary services they provide. In
theory, the service flow is given by the utility level. Consequently the
marginal services flow of a monetary asset is its marginal utility. In
equilibrium, the marginal service flow of a monetary asset is
proportional to its rental rate, so the change in the value of a
monetary asset's service flow per dollar of the asset held can be
approximated by its user cost. The marginal monetary services of the
components of Divisia aggregates are likewise proxied by the user costs
of the components assets. The user cost of each component is
proportional to the interest income foregone by holding it rather than a
pure store-of-wealth asset - an asset that yields a high rate of return
but provides no monetary services. Currency and non-interest-bearing
demand deposits have the higher user cost because they earn no explicit
interest income. Consequently they get the largest weights in the
Divisia measure. On the other hand, pure store-of-wealth assets get zero
weights.(7)

The object of a Divisia measure is to construct an index of the
flow of monetary services from a group of monetary assets, where the
monetary service flow per dollar of the asset held can vary from asset
to assert.(8) Applying an appropriate index number to a group of assets
is not sufficient, however, to get a correct measure of the flow of
monetary services. The index must also be constructed from a assets that
can be aggregated under conditions set by consumer demand theory. The
objective of economic aggregation is to identify a group of goods that
behave as if they were a single commodity. A necessary condition for
this is block-wise weak separability. Block-wise weak separability
requires that consumers' decisions about goods that are outside the
group do not influence their preferences over the goods in the group
whatsoever.(9) If this condition is satisfied, consumers behave just as
though they were allocating their incomes over a single aggregate
measure of monetary services and all other commodities to maximize their
utility. Their total expenditure on monetary services is subsequently
allocated over the various financial assets that provide such services.

The Divisia index generates such a monetary aggregate. Moreover, in
continuous time it has been shown to be consistent with any unknown
utility function implied by the data. In discrete time the Divisia index
is in the class of superlative index numbers. Simple-sum indexes, on the
other hand, do not have this desirable property. Thus they have no basis
in their consumer demand theory or aggregation theory.(10)

In principle, all financial assets other than pure store-of-wealth
assets provide some monetary services. Which assets can be combined into
a meaningful monetary aggregate is in empirical issue because economic
theory does not tell us which group of assets satisfies the condition of
block-wise weak separability. Unfortunately, the most widely used test
for weak separability is not powerful.(11) Consequently, it has been
common simply to create Divisia indexes under the maintained hypothesis
that the assets that compose the aggregate satisfy this condition. Thus
the issues of the appropriate method of aggregation and the appropriate
aggregate have been treated seperately.(12)

SIMPLE-SUM AND DIVISIA

MONETARY INDEXES

A simple-sum monetary aggregate is a measure of the stock of
financial assets that compose it, whereas a Divisia monetary aggregate
is a measure of the flow of monetary services from the stocks of
financial assets that compose it.(13) For this reason alone, the methods
of measurement are quite different. Simple-sum aggregates are obtained
by simply adding the dollar amounts of the component assets. On the
other hand, Divisia monetary aggregates are obtained by multiplying each
component asset's growth rate by its share weights and adding the
products. A component's share weight depends on the user costs and
the quantities of all component assets.(14) Specifically, the share
weight given to the [j.sup.th] component asset at time t is its share of
total expenditures on monetary services; that is,

[Mathematical Expression Omitted]

where [q.sub.j] denotes the nominal quantity of the [j.sup.th]
component assets, [u.sub.j] denotes the [j.sup.th] component's user
cost and n donotes the number of component financial assets. The user
cost is equal to [(R-r.sub.j)]p/ (1 + R), where R is the benchmark rate
(that is, the rate on the pure store-of-wealth asset), [r.sub.j], is the
own rate on the [j.sup.th] component, and p is the true cost-of-living
price index that cancels out of the numerator and denominator of the
shares. The growth rate of the [i.sup.th] Divisia monetary aggregate,
[GDM.sub.i], is given by

[Mathematical Expression Omitted]

where [g.sub.jt] is the growth rate of [q.sub.jt].(15)

A Comparison of Simple-Sum and

Divisia Monetary Aggregates

Because the Divisia aggregates are an alternative to the
conventional simple-sum aggregates, it is instructive to compare them.
When constructing data in this section, the authors used an extension of
the Farr and Johnson (1985) method. The Appendix present details of the
construction of the Divisia monetary aggregates used here.

A Divisia monetary index is an approximation to a nonlinear utility
function. Because it is an index, the level of utility is an arbitrary
unit of measure; the level of the index has no particular meaning.(16)
Nevertheless, because they are alternative measures of money, the
Divisia and simple-sum aggregates are frequently compared to see how any
analysis of the effects of monetary policy or other issues might be
affected by the method of aggregation. The comparison of the levels of
the simple-sum and Divisia measures is made by normalizing both measures
so that they equal 100 at some point in the series, usually the first
observation.(17) Comparisons of the levels and growth rates of the
Divisia and simple-sum measures are presented in figures 1-5 for four
monetary aggregates, M1A, M1, M2 and M3, and for total liquid assets,
L.(18) The figures have two scales. The left-hand scale indicates the
growth rate, and the right-hand scale indicates the level of the series.
Both indexes equal 100 in January 1960.

M1A

M1A comprises currency and non-interest-bearing demand deposits
held by households and businesses. Although neither household nor
business demand deposits earn explicit interest, business demand
deposits are assummed to earn an implicit own rate of return
proportional to the rate paid on one-month commercial paper.(19)
Consequently, additional units of business demand deposits are assumed
to yield a smaller flow of monetary services than are additional units
of household demand deposits. On the other hand, the simple-sum measure
implicitly assumes that each unit of each component provides the same
flow of monetary services. Hence the Divisia aggregate gives more weight
to the growth rates of currency and household demand deposits than does
the simple-sum aggregate.(20)

The average differences in the growth rates of the simple-sum and
Divisia measures of M1A for the entire sample period, January 1960 to
December 1992, and for selected sub-periods are presented in table 1.
Because currency generally grew more rapidly than demand deposits over
the sample period, the growth rate of Divisia M1A averaged about half a
percentage point higher than the growth rate of simple-sum M1A over the
entire period.(21) Much of this differences occurs during the latter
part of the 1980s, when the growth rate of demand deposits generally
slowed relative to the growth rate of currency.(22) This more rapid
growth of the Divisia measure is reflected in a generally widening gap
between the levels of the indexes.

M1

The behavior of simple-sum and Divisia M1 is similar to that of
M1A. Indeed, the growth rates of simple-sum and Divisia M1 were similar
until the late 1970's, when the growth of interest-bearing NOW
accounts began to accelerate. The sharp rise in NOW accounts after their
nationwide introduction on January 1, 1981, tended to increase the
growth rate of the simple-sum measure relative to the Divisia measure
because the growth rate of NOW accounts gets a smaller weight in the
Divisia measure. As a result, the Divisia measure grew more slowly on
average than the simple-sum measure from the late 1970s until the
mid-1980s, after growing more rapidly previously. However, in neither
period is the average difference in the growth rate of the alternative
measures large.(23)

After the late 1980s the Divisia measure grew more rapidly than the
simple-sum measure, reflecting the rise in the growth rate of currency
relative to the growth rate of checkable deposits. Of course, the
smaller average difference in the growth rates of the alternative M1
aggregates compared with 1 M1A is reflected in a smaller difference in
the levels of the two indexes as well.

M2, M3 and L

Not surprisingly, larger differences arise when the monetary
measures are broadened to include savings-type deposits because their
explicit own rates of return are higher than those of transactions
deposits. The higher own rate reduces the share weights of these
components assets relative to the weights they receive in the simple-sum
measures. During the sample period the growth rates of the broader
simple-sum aggregates tend to be substantially larger than those of the
corresponding Divisia measures. For the broader measures, the average
growth rates of the simple-sum measures are about 2 percentage points
greater than the corresponding Divisia measures over the entire sample
period.

Much of this differences arises from the late 1970s to the
mid-1980s and is likely due to financial innovation and deregulation in
the period. The late 1970s witnessed a marked acceleration in the growth
of money market mutual funds. These accounts paid relatively high
interest rates and had limited transactions capabilities. A number of
new deposit instruments that paid higher market interest rates were
introduced in the early 1980s and Regulation Q interest rate ceiling
were being phased out.(24) Moreover, short-term interest rates reached
very high levels in the early 1980s. With share weights sensitive to the
spread between an asset's own rate of return and the return on the
benchmark assets, it is not surprising that the Divisia measures grew
markedly slower than the corresponding simple-sum measures during this
period. Nevertheless, the significantly slower growth of the broader
Divisia measures during this period is more consistent with the
disinflation of the period than is the growth of the simple-sum
aggregates, whose growth remained fairly rapid. Although the growth
rates of the broader Divisia and simple-sum aggregates have been
essentially the same, on average, since a about the mid-1980s, the
pattern of growth of these alte alternative measures is somewhat
different.

A Comparison of Broader Divisia

Aggregates

That Divisia aggregation gives relatively small weight to less
liquid assets that yield high rates of return suggests that differences
in the growth rates of successively broader Divisia monetary aggregates
will tend to get smaller.(25) The levels of Divisia M2, M3 and L
presented in figure 6 and simple correlations of the compounded

Gannual growth rates of these Divisia aggregates presented in table 2
confirm this. The growth rates of Divisia M3 and L differ little from
the growth rate in Divisia M2. This implies that adding successively
less liquid assets to those in M2 adds little to the flow monetary
services.(26) That the average difference in the growth rates of Divisia
M2 and L is nearly zero over the entire sample period is reflected by
the levels of the two Divisia aggregates, which are essentially equal by
the end of the sample. Divisia M3, however, has, grown more rapidly than
the other measures, so the spread between its level and the levels of
Divisia M2 and L has widened over the sample period.

CONCLUDING REMARKS

Despite their theoretical advantage, Divisia and other weighted
monetary aggregates have garnered relatively little attention outside of
academe, and the official U.S. monetary aggregates remain simple-sum
aggregates. The official reliance on simple-sum aggregates will probably
continue unless the Divisia aggregates or other alternative weighted
aggregates are shown to be superior in economic and policy analysis.
Although nothing definitive can be said about this issue from the simple
analysis of the data presented here, a few observations are offered.

First, that the growth rates of the narrow simple-sum sum and
Divisia monetary aggregates are quite similar suggests that the method
of aggregation may not be important at low levels of aggregation.(27)
For example, it does not appear that conclusions about the long-run
effects of money growth on inflation would be much different using
either simple-sum or Divisia M1 or M1A. The average difference in the
growth rates of narrow simple-sum and Divisia monetary aggregates is
small. This observation is consistent with the empirical work of
Barnett, Offenbacher and Spindt (1984) who, using a broad array of
criteria, found that the difference in the performance of simple-sum and
Divisia monetary aggregates was small at low levels of aggregation.

Second, the method of aggregation is likely to be more important
for broader monetary aggregates. Beyond some point, however, a further
broadening of the monetary aggregate makes little difference. For the
United States, the differences in the average growth of Divisia M2, M3
and L are small. Consequently, long-run analysis using the growth rates
of any of these Divisia aggregates is likely to produce similar results.
Monthly growth rates of these Divisia aggregates are also highly
correlated. Hence it would not be too surprising to find the broader
Divisia aggregates perform similarly to one another in many short-run
analyses as well.

These observations point to the critical need for more work to
determine which financial assets should be included in the appropriate
monetary aggregate. In consumer demand theory, these assets must satisfy
the condition of weak separability. If analysis suggests a relatively
narrow monetary aggregate such as M1, policymakers may be reluctant to
adopt the theoretically superior index measure because, as a practical
matter, the method of aggregation may not be empirically important.

If such test point to an aggregate that includes a much broader
array of financial assets, the practical case for the weighted
aggregates will be enhanced. Even casual analysis of simple-sum and
Divisia monetary aggregate data show differences in both the levels and
grown rates of these aggregates that are large, suggesting that the
method of aggregation is important. Consequently, the method of
aggregation should also be a concern for those who favor broader
monetary aggregates on other grounds. The objective of the present
article in publishing Divisia monetary statistics is to stimulate
further empirical research both on the importance of monetary
aggregation and on the role money in the economy.

(1) The discussion in this section is based on consumer demand
theory. This may not be a serious limitation. For example, Feenstra
(1986) has shown that money in the utility function is equivalent to
other approaches. These approaches assume, however, that all of the cost
and benefits of money are internalized, and it is commonly believed that
there are externalities to the use of money in exchange (see Laidler
[1990]). (2) This need to be true for the economy as a whole when
measured over a sufficiently long time interval. In this case the amount
of each from of money multiplied by its turnover velocity will equal
total expenditures. This is the basis for the velocity of the demand for
money. Fisher (1911) explicitly recognized that turnover velocities of
currency and checkable deposits would likely be different. He
circumvented this problem by assuming that there was an optimal
currency-to-deposit ratio that would be a function of economic
variables. Given these variables, the demand for the two alternative
monetary assets was taken to be strictly proportional. Moreover, because
individuals were free to adjust their money holdings between currency
checkable deposits quickly and at low cost, Fisher argued that the
actual ratio would deviate from the desired ratio for only short
periods. For some recent evedince that the actual currency-to-deposit
ratio might be determined by the policy actions of the Federal Reserve,
see Garfinkel and Thornton (1991). The possibility that currency and
checkable deposits have different turnover velocities is the basis for
Spindt's (1985) weighted monetary aggregate, MQ.

(3) There is an issue of whether the fixed ratio was endogenous from
either the perspective of supply or demand, or the result of arbitrary
legal restrictions. From the demand, side this would require that these
assets be perfect substitutes for all transactions. From the supply
side, Pesek and Saving (1967) argued that the one-to-one exchange rate
was a natural outcome of competitive pressures in the banking industry.
Whether the fixed one-to-one is the endogenous outcome of a free market
economy or is simply due to legal restrictions remains controversial.

Of course today some checkable deposits earn explicit interest.
Consequently such deposits are a better store of wealth than currency.
They are also a preferable medium of exchange for some, but not all,
transactions.

(4) There has been a difference of opinion about the degree of
emphasis that should be placed on the asset and transactions motives for
holding money. Indeed, Laidler (1990, pp. 105-6) has noted that " .
. .the most extraordinary development in monetary theory over the past
fifty year is they way in which money's means-of-exchange and
unit-of-account roles have vanished from what is widely regarded as the
mainstream of monetary theory."

Broaching the medium-of-exchange line of demarcation between money
and non-money assets also gave rise to an extensive literature on the
empirical definition of money. For a critique of this literature and the
idea of distinguishing between monetary and non-monetary assets based on
the concept of the temporary abode of purchasing power, see Mason
(1976).

(5) At one point the Federal Reserve published data on five
alternative monetary aggregates. (6) Chetty's work was motivated by
the Gurley/Shaw hypothesis and the general lack of agreement in the
empiricial findings of Feige (1964) and others about the degree of
substitutability between money and near-money assets. Gurley and Shaw
(1960) suggested that the effectiveness of monetary policy was limited
because of the high degree of substitutability between money (currency
and demand deposits) and near-money (various bank and nonbank
savings-type accounts) assets. Subsequent research has tended to support
Feige's finding of a relatively low degree of substitutability
between transactions media and liquid, non-medium-of-exchange assets.
See Fisher (1989) for a survey of much of this literature.

(7) There does not appear to be agreement about what constitutes the
best proxy measure for the theoretical pure store-of-wealth asset.
Barnett, Fisher and Serletis (1992, p. 2,093) state the following,
"The benchmark asset is specifically assumed to provide no
liquidity or other monetary services and is held solely to transfer
wealth intertemporally. In theory R (the benchmark rate) is the maximum
expected holding period yield in the economy. It is usually defined in
practice in such a way that the user costs for the monetary assets are
(always) positive." Parentheses added. The Baa bond rate, or the
highest rate paid on any of the component assets when the yield curve
becomes inverted, has frequently been used to construct Divisia
aggregates.

(8) See Barnett, Fisher and Serletis (1992) and Yue (1991a and b) for
more detailed analyses of issues in monetary aggregation.

(9) Technically the marginal rates of substitution between any two
goods inside the group must be independent of the quantities of the
goods consumed that are outside of the group.

(10) Fisher (1922) was especially critical of the simple-sum index in
his extensive analysis of index numbers. In particular, Fisher argued
that simple-sum aggregates cannot internalize pure substitution effects
associated with relative price changes. Thus changes in utility, which
should occur only as a result of the income effect associated with
relative price changes, occur in simple-sum aggregates because of both
income and substitution effects.

(11) The most widely used test, developed by Varian (1982, 1983), is
not statistical. The null hypothesis of weak separability is rejected if
a single violation of the so-called regularity conditions is found.
Because tests for weak separability lack power, Barnett, Fisher and
Serletis (1992, p. 2,095) argue that "existing methods of
conducting such tests are not . . . very effective tools of
analysis," See Barnett and Choi (1989) for evidence indicating that
available tests of block-wise weak separability are not very dependable.
For results of tests for weak separability, see Belongia and Chalfant
(1989) and Swofford and Whitney (1986), 1987).

(12) A common practice both in the United States and abroad is to
construct Divisia monetary aggregates for collections of assets that are
reported by the country's central bank. For example, see Yue and
Fluri (1991), Belongia and Chrystal (1991) Ishida (1984).

(13) It should be noted that the accounting stock that is, the sum of
the dollar amounts of all assets that are considered money, is not
necessrily equal to the capital stock of money. The accounting stock is
the present value of both service flow of money and the interest income
(the service as a store of value). The economic capital stock of money
comprises only the present value of the flow of monetary services. See
Barnett (1991) for the formula for the economic capital stock of money.

(14)For the Divisia monetary aggregates, the share weight of each
component's growth rate is its expenditure share of total
expenditures on monetary services. Theoritically the share weights for
the Divisia monetary aggregates are not a function of prices or user
costs, but of quantities. The observable user costs are substituted for
the unobservable marginal utilities under the implicit assumption of
market-clearing equilibrium, where each consumer holds an optimal
porfolio of monetary and nonmonetary assets. For the simple-sum monetary
aggregates, the share weights are the components' share of the
aggregate.

(15) GDM[sub.j] - In Dl[sub.t-1,] where Dl denotes the Divisia index.
The index is initialized at 100, that is, Dl[sub.0]=100. See Farr and
Johnson (1985) for more details.

(16) Rotemberg (1991) derives a weighted monetary aggregate stock
under the conditions of risk neutrality and stationarity expectations;
however, Barnett (1991) shows that this measure is the discounted value
of future Divisia monetary service flows.

(17) An alternative justification for comparing the Divisia and
simple-sum aggregates might come from noting that the appropriate
Divisia monetary aggregate would be the simple-sum aggregate if all of
the component assets had identical own rates. Such a comparison is
tenuous, however, because the actual level of the simple-sum aggregate
might have been different from the observed level had the user costs
actually equal.

It is common to compare the levels and growth rates of simple-sum
and Divisia monetary aggregates. For example, see Barnett, Fisher and
Serletis (1992). Because Divisia indexes involve logarithms, the growth
rate of a component asset is plus or minus infinity, respectively, when
a component is introduced or eliminated. To circumvent this problem the
Divisia index is replaced by Fisher's ideal index at these times
and the user cost is measured by its reservation price during the period
that precedes the introduction or follows the elimination of the asset.
See Farr and Johnson (1985) for a discussion of this procedure.

(18) Note that the simple-sum aggregates presented here are not
identical to the official published series. The official series are
obtained by adding the non-seasonally adjusted components and seasonally
adjusting the aggregate as a whole or by adding large subgroups of
component asset that have been seasonally adjusted as a whole. The
simple-sum aggregate presented here are obtained by adding the
components after each component (that has a distinctive seasonal) has
been seasonally adjusted. See the Appendix for details. A comparison of
the series used here and the official series shows that the differences
are small.

(19) Alternatively, estimates of the own rate on household demand
deposits could also be used. However, such a series was not available
for the entire sample period. Moreover, the desire was to follow the
procedure used by Farr and Johnson (1985) as closely as possible.

(20) In both cases, the sum of the weights must equal unity.

(21) Currency grew at an annual rate of 7 percent during the entire
period, whereas household and business demand deposits both grew at a
3.2 percent annual rate.

(22) This is a period of very reserve growth. Because reserves and
checkable deposits are tied closely together under the present system of
reserve requirements it is not surprising that is also a period of slow
growth in checkable deposits, including household and business demand
deposits. See Garfinkel and Thornton (1991) for a discussion of the
relationship between reserves and chekable deposits under the present
system of reserve requirements.

(23) We have refrained from using the phrase "statistically
significant" because these observations are clearly distributed
identically and independently, so the "t-statistics" reported
in table 1 are biased and neither the direction nor extent of the bias
is known. These statistics are presented to give the reader a rough
approximation of the magnitude of the differences in the growth rates.
Correlegrams of the differences in the growth rates of simple-sum and
Divisia M1A and M1 show some lower level persistence through the sample
period and some large spikes at seasonal frequencies after 1969.
Correlegrams for the difference in the growth rates of the broader
monetary aggregates reveal some higher level persistence. In any event,
differences that are small in absolute value tend to be small relative
to the estimated standard errors, and differences that are large in
absolute value tend to be large in relative terms.

Another measure of the distance between the growth rates is the
square root of the sum of the squared differences in the growth rates.
These measures for the entire sample period are 58.5, 52.1, 69, 81.4 and
77.6 for M1A, M1, M2, M3 and L, respectively. These data are broadly
comparable with those presented in table. 1.

(24) For a discussion of the financial innvations of this period see
Gilbert (1986) and Stone and Thornton (1991).

(25) Of course this tendency also exists for the simple-sum
aggregates. For the simple-sum aggregate, the growth rate of each
component is weighted by the component's share of the total asset.
Hence the growth rates of successively broader monetary aggregates could
diverge if the marginal components were successively larger. For
example, this is what happens from M1 to M2. The growth rates tend to
converge, however, because the marginal components are smaller. This
tendency is exacerbated in the Divisia measures because of smaller
weights associated with higher own rates of return on successively less
liquid assets.

(26) The average differences in the growth rates of Divisia M2, M3
and L over the sample period are small (less than 0.12 percentage points
in absolute value). The absolute values of the average differences in
the growth rates of simplesum M2, M3 and L are larger than those of the
corresponding Divisia measures; the standard errors are also much
larger.

(27) There may be some differences in the levels, however, because
the levels of the simple-sum and Divisia measures do not appear to be
cointegrated at any level of aggregation.

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